Visualizing Complex Numbers

Date: 03/19/2001 at 22:30:39
From: Anonymous
Subject: Complex Numbers
What exactly are imaginary numbers and how are they used? I do not
quite understand how the square root of -1 is possible.
Thank you.

Date: 03/20/2001 at 09:23:20
From: Doctor Floor
Subject: Re: Complex Numbers
Hi,
Thanks for writing.
This is a difficult question!
Imaginary numbers are all numbers that are the product of a real
number and the square root of -1. The set of complex numbers is the
set of numbers that can be found as the sum of a real and an imaginary
number.
First of all, to understand complex numbers we have to get rid of the
number line alone. Instead we consider a number plane, with a grid.
The usual real numbers are found along the x-axis, so that's our usual
number line. So the point (x,0) in the plane we take for the real
number x.
To find the square root of -1, we first try to find out what
multiplying by the square root of -1 should do. We do that by
geometric interpretation. By such interpretation, multiplying by -1 is
the same as rotating about the origin through 180 (or -180) degrees.
Multiplying by the square root of -1 should do this halfway, so that
after doing it twice it would be the same as multiplying by -1. And
halfway rotating through 180 (or -180) degrees, is not difficult:
that's the same as rotating through 90 (or -90) degrees.
So we can take rotating through 90 degrees as "multiplying by the
square root of -1" (through -90 degrees is then muliplying by minus
the square root of -1).
Now, the square root of -1 as a number in the plane is found as 1
multiplied by the square root of -1, or (1,0) after rotation through
90 degrees about (0,0), i.e. the point (0,1).
And we see that imaginary numbers will all be found on the y-axis.
Therefore, when we talk about the plane of complex numbers, the x-axis
is called the real axis, and the y-axis the imaginary axis.
I hope this helps you.
Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/